diff options
Diffstat (limited to 'graphics/pstricks/contrib/pst-func/doc')
-rw-r--r-- | graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib | 8 | ||||
-rw-r--r-- | graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf | bin | 4050523 -> 4065407 bytes | |||
-rw-r--r-- | graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex | 107 |
3 files changed, 102 insertions, 13 deletions
diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib index e749f71bfb..89f7eaad31 100644 --- a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib +++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib @@ -166,3 +166,11 @@ year = {2012}, url = {/graphics/pstricks/contrib/pst-tools} } + +@online{weisstein, + author = {Weisstein, Eric W.}, + title = {Noncentral Student's t-Distribution}, + subtitle = {From MathWorld--A Wolfram Web Resource}, + url = {https://mathworld.wolfram.com/NoncentralStudentst-Distribution.html}, + note = {Probability and Statistics; Statistical Distributions; Continuous Distributions } + } diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf Binary files differindex b791d5e480..62249b7e51 100644 --- a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf +++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex index 8e87d3a6d3..b3f0c2ecae 100644 --- a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex +++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex @@ -1,6 +1,6 @@ \RequirePackage{pdfmanagement-testphase} \DeclareDocumentMetadata{} -%% $Id: pst-func-doc.tex 306 2021-10-29 14:48:08Z herbert $ +%% $Id: pst-func-doc.tex 691 2023-04-03 10:51:04Z herbert $ \documentclass[fontsize=11pt,english,BCOR=10mm,DIV=12,bibliography=totoc,parskip=false,headings=small, headinclude=false,footinclude=false,oneside]{pst-doc} \usepackage{pst-func} @@ -1158,7 +1158,7 @@ f(t) = constI \cdot J_n + constII where \Lkeyword{constI} and \Lkeyword{constII} must be real PostScript expressions, e.g. \begin{lstlisting}[style=syntax] -\psset{constI=2.3,constII=t k sin 1.2 mul 0.37 add} +\psset{constI=2.3,constII=t toDegrees sin 1.2 mul 0.37 add} \end{lstlisting} The Bessel function is plotted with the parametricplot macro, this is the @@ -1174,13 +1174,11 @@ In particular, note that the default for time consuming at this setting, it can be decreased in the usual way, at the cost of some reduction in graphics resolution. -\begin{LTXexample} -{ +\begin{LTXexample}[pos=t] \psset{xunit=0.25,yunit=5} \begin{pspicture}(-13,-.85)(13,1.25) \rput(13,0.8){% - $\displaystyle J_n(x)=\frac{1}{\pi}\int_0^\pi\cos(x\sin t-nt)\dt$% -} + $\displaystyle J_n(x)=\frac{1}{\pi}\int_0^\pi\cos(x\sin t-nt)\dt$} \psaxes[Dy=0.2,Dx=4]{->}(0,0)(-30,-.8)(30,1.2) \psset{linewidth=1pt} \psBessel[linecolor=red]{0}{-28}{28}% @@ -1188,23 +1186,21 @@ way, at the cost of some reduction in graphics resolution. \psBessel[linecolor=green]{2}{-28}{28}% \psBessel[linecolor=magenta]{3}{-28}{28}% \end{pspicture} -} \end{LTXexample} -\begin{LTXexample} -{ + +\begin{LTXexample}[pos=t] \psset{xunit=0.25,yunit=2.5} \begin{pspicture}(-13,-1.5)(13,3) \rput(13,0.8){% - $\displaystyle f(t) = 2.3 \cdot J_0 + 1.2\cdot \sin t + 0.37$% -} + $\displaystyle f(t) = 2.3 \cdot J_0 + 1.2\cdot \sin t + 0.37$} \psaxes[Dy=0.8,dy=2cm,Dx=4]{->}(0,0)(-30,-1.5)(30,3) \psset{linewidth=1pt} -\psBessel[linecolor=red,constI=2.3,constII={t k sin 1.2 mul 0.37 add}]{0}{-28}{28}% +\psBessel[linecolor=red,constI=2.3,constII={t toDegrees sin 1.2 mul 0.37 add}]{0}{-28}{28}% \end{pspicture} -} \end{LTXexample} + %\clearpage \subsection{Modfied Bessel function of first order} The modified Bessel function of first order is defined as @@ -1216,6 +1212,7 @@ I_\nu(x) &= \left(\frac12 x\right)^\nu % \noindent The syntax of the macro is + \begin{BDef} \Lcs{psModBessel}\OptArgs\Largb{xStart}\Largb{xEnd} \end{BDef} @@ -1223,6 +1220,11 @@ I_\nu(x) &= \left(\frac12 x\right)^\nu The only valid optional argument for the function is \Lkeyword{nue}, which is preset to 0, it shows $I_0$. + + + + + \begin{LTXexample} \begin{pspicture}(0,-0.5)(5,5) \psaxes[ticksize=-5pt 0]{->}(5,5) @@ -1909,6 +1911,70 @@ and the following syntax %The $t_\nu$-distribution has mode 0. +\subsection{Student's noncentral $t$-distribution} + + +A generalization of Student's t-distribution known as the noncentral Student's t-distribution is given by + +\[ + P(x)=\frac{\nu^{\nu/2}\nu!}{2^{\nu}e^{\lambda^2/2}(\nu+x^2)^{\nu/2}\Gamma(\frac12\nu)} + \left\{ + \frac{\sqrt{2}\lambda x\, {}_1F_1\left(\frac12 \nu+1;\, \frac32; \,\frac{\lambda^2x^2}{2(\nu+x^2)}\right)} + {(\nu+x^2)\Gamma[\frac12(\nu+1)]} + + \frac{{}_1F_1\left(\frac12(\nu+1);\, \frac12; \, \frac{\lambda^2x^2}{2(\nu+x^2)}\right)}{\sqrt{\nu+x^2}\Gamma(\frac12\nu+1)}\right\} +\] + +where $\Gamma(z)$ is the gamma function and ${}_1F_1(a;b;z)$ is a confluent hypergeometric function of the first kind.~\parencite{weisstein} + + +\[ {}_1F_1(a;b;z) = 1+\frac ab z + \frac{a(a+1)}{b(b+1)} \frac{z^2}{2!}+ \cdots=\sum_{k=0}^\infty\frac{(a)_k}{(b)_k} \frac{z^k}{k!} \] + + +%mu = lambdasqrt(n/2)(Gamma(1/2(n-1)))/(Gamma(1/2n)) +%sigma^2 = ((lambda^2+1)n)/(n-2)-(lambda^2n[Gamma(1/2(n-1))]^2)/(2[Gamma(1/2n)]^2). + +The macro has the same syntax as the centered Student's distribution: + +\begin{BDef} +\Lcs{psNonCentralTDist}\OptArgs\Largb{x0}\Largb{x1} +\end{BDef} + + +There exists an additional optional parameter \Lkeyword{lambda}, which is preset to 0.5. + + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=0.75cm,yunit=10cm} +\begin{pspicture}(-6,-0.1)(11,.5) + \psaxes[Dy=0.1]{->}(0,0)(-4.5,0)(10,0.5) + \psset{linewidth=1pt,plotpoints=100} + \psNonCentralTDist[linecolor=red,lambda=0,nue=1]{-5}{10}% same as \psTDist + \psNonCentralTDist[linecolor=green,lambda=0,nue=4]{-5}{10}% dito + \psNonCentralTDist[linecolor=red,lambda=2,nue=1]{-5}{10} + \psNonCentralTDist[linecolor=green,lambda=2,nue=4]{-5}{10} +\end{pspicture} +\end{LTXexample} + + + + + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1.25cm,yunit=10cm} +\begin{pspicture}(-6,-0.1)(6,.5) + \psaxes[Dy=0.1]{->}(0,0)(-4.5,0)(5.5,0.5) + \psset{linewidth=1pt,plotpoints=100} + \psTDist[linecolor=blue]{-4}{4} + \psNonCentralTDist[linecolor=red]{-4}{5} + \psNonCentralTDist[linecolor=green,nue=4,lambda=2]{-4}{5} +\end{pspicture} +\end{LTXexample} + + + + + + \clearpage \subsection{$F$-distribution} A continuous statistical distribution which arises in the testing of @@ -2173,6 +2239,21 @@ origin of the coordinate system. \endpsgraph \end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{lly=-13mm,llx=-5mm} +\psgraph[Dx=0.2,Dy=0.2,axesstyle=frame](0,0)(1,1){6cm}{6cm} +\psline[linestyle=dashed](1,1) +\psLorenz[Gini,linecolor=red,linewidth=1.5pt]<.2 .2 .2 .2 .2>{.1 .1 .1 .35 .35 } +\psLorenz[Gini,linecolor=blue,linestyle=dashed,linewidth=1.5pt]{.1 .1 .1 .35 .35 } +\endpsgraph +\end{LTXexample} + + + + + + \clearpage \subsection{\nxLcs{psLame} -- Lam\'e Curve, a superellipse} A superellipse is a curve with Cartesian equation |